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Theorem ovresd 7573
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1 (𝜑𝐴𝑋)
ovresd.2 (𝜑𝐵𝑋)
Assertion
Ref Expression
ovresd (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2 (𝜑𝐴𝑋)
2 ovresd.2 . 2 (𝜑𝐵𝑋)
3 ovres 7572 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
41, 2, 3syl2anc 584 1 (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106   × cxp 5674  cres 5678  (class class class)co 7408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688  df-iota 6495  df-fv 6551  df-ov 7411
This theorem is referenced by:  sscres  17769  fullsubc  17799  fullresc  17800  funcres2c  17851  psmetres2  23819  xmetres2  23866  prdsdsf  23872  xpsdsval  23886  xmssym  23970  xmstri2  23971  mstri2  23972  xmstri  23973  mstri  23974  xmstri3  23975  mstri3  23976  msrtri  23977  tmsxpsval  24046  ngptgp  24144  nlmvscn  24203  nrginvrcn  24208  nghmcn  24261  cnmpt1ds  24357  cnmpt2ds  24358  ipcn  24762  caussi  24813  causs  24814  minveclem2  24942  minveclem3b  24944  minveclem3  24945  minveclem4  24948  minveclem6  24950  ftc1lem6  25557  ulmdvlem1  25911  abelth  25952  cxpcn3  26253  rlimcnp  26467  hhssnv  30512  madjusmdetlem3  32804  qqhcn  32966  qqhucn  32967  ftc1cnnc  36555  ismtyres  36671  isdrngo2  36821  naddcnffo  42104  rngchom  46855  ringchom  46901  irinitoringc  46957  rhmsubclem4  46977
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